My money is that *some* parts of it (esp. the parts nearest the coop and in any other attractive locations such as under trees or bushes) will rather rapidly become bare dirt, and the rest will gradually follow, probably via a phase where it's more weeds than grass

Maybe you could split it in half. Cross fence it. Let them destroy one side while the other grows then rotate them. Rotational grazing works wonders for livestock I do not see any reason the principle cant be applied to chickens in that much space. Since the divider is not for predator protection chicken wire would be adequate. Edited too add you may have to reseed from time to time. Deer plot mixes in 50 pound bags are cheaper than grass seed and better for the chickens.

I saw a nice coop setup once that had a run on both sides. The person who owned it would use one run for chickens the other for crops. They changed sides every year so they always had fertile growing space and a clean run. If i had room on the other side of my barn that is what I would do.

There was an idea on here last year about building frames and putting hardware cloth on top of it to protect the grass from their digging. Not sure if anyone has tried it but maybe put in a few places around the run you could still have some grass for them.